The three options are:
LAN. This option lets you play over your local area network, so you and other people can play with each other at home (provided you have a console for everyone). LAN is also useful for doing multiplayer with yourself and testing out pure solo characters, as the strength of monsters is gagued to the number of players you have in the game and if there are no other computers on your network, you won't be disturbed at inopportune times (as can occasionally happen with internet play). Characters you build are stored on your local computer and can be transferred between LAN and internet options.
Internet. This option hooks you up to the internet to play online. You may use your LAN characters for internet play as well, as both are stored on your home computer.
GameSpy. This hooks you up to the gamespy network to play online. Gamespy characters are limited to gamespy only, as they are stored on the gamespy servers. This has the benefit of you being bale to access that character from any computer anywhere in the world, but a few people have had problems with world saves not being stored on the gamespy servers (and losing their progress through quests when switching to another computer).
As far as the inn bug goes, I've experienced this in BW multiplayer. GPG really needs to patch the accursed thing.
From cradle to coffin... Shall my wickedness... Be your passion